I'm having trouble computing the integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$
I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very effective.

Write the numerator (here $\sin x$) as a linear combination of the denominator and the derivative of the denominator:
$$A(\sin x+ \cos x) + B( \cos x- \sin x) = \sin x$$
Solve for $A$ and $B$ and split the fraction accordingly. Integrating give a linear term and an $\ln$